# [seqfan] Re: An arithmetic conjecture

Martin Fuller martin_n_fuller at btinternet.com
Tue Mar 24 18:46:07 CET 2009

```David Wilson,

I would like to publish, but I do not know how or where.  Would you be interested in co-authoring?  I could do most of the writing and type-setting, and you would provide advice and direction.

Martin Fuller

--- On Fri, 20/3/09, David Wilson <dwilson at gambitcomm.com> wrote:

> From: David Wilson <dwilson at gambitcomm.com>
> Subject: [seqfan] Re: An arithmetic conjecture
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Friday, 20 March, 2009, 9:21 PM
> No, I think you're right.
>
> I was mistaken that the number of nth base-b zeroless
> number was
> approximately exponential in n (c^n), when it is actually
> polynomial in
> n (n^c). That makes all the difference.
>
> I redid the math, and your k value dropped out of it.
>
> So I rescind my conjecture about k-automatic sequences in
> deferences to
> your work. You really should publish.
>
> I'll take a look at palindromes when I can.
>
> Martin Fuller wrote:
> >> From: David Wilson <dwilson at gambitcomm.com>
> >> Subject: [seqfan] Re: An arithmetic conjecture
> >> To: "Sequence Fanatics Discussion list"
> <seqfan at list.seqfan.eu>
> >> Date: Tuesday, 17 March, 2009, 2:46 PM
> >>
> > [cut]
> >
> >> I have a conjecture along these lines:
> >>
> >> If two bases a >= 2 and b >= 2 and two sets
> of
> >> integers A and B where
> >>
> >>     a and b are not powers of the same integer
> (e.g, a = 4,
> >> b = 8 is
> >> unacceptable)
> >>     A and B are infinite,
> >>     A and B have limit density 0 over the
> integers,
> >>     A is a-automatic (the base-a representations
> of the
> >> elements of A
> >> form a regular language) and B is b-automatic.
> >>
> >> Then A and B have finite intersection.
> >>
> >> Example:
> >>
> >> Let a = 2, b = 10, A = powers of 2, B = numbers
> with no 0
> >> in their
> >> base-10 numerals.
> >>
> >> This example easily conforms to the first three
> conditions.
> >> A is 2-automatic, with its base-2 numerals forming
> the
> >> regular language 10*.
> >> B is 10-automatic, with is base-10 numerals
> forming the
> >> regular language
> >> [123456789]+
> >>
> >> My conjecture implies that A and B have finite
> >> intersection, that is,
> >> there are a finite number of powers of 2 without
> zeroes in
> >> their base-10
> >> representations.
> >>
> >> My conjecture also implies your conjecture.
> >>
> >>
> > [cut]
> >
> > Some counter-conjectures (using your reasoning from 7
> March 2009):
> >
> > A,B = numbers without a zero in base a,b
> > I conjecture that the intersection is infinite for any
> pair a,b >= 3
> > Example: a=3, b=4
> > The sequence starts 1, 2, 5, 7, 13, 14, 22, 23, 25,
> 26, 41, 43, 53, 121, 122, 125, 149, 151, 157, 158, 214, 215,
> 229, 230, 233, 238, 239, 365, 367, 373, 374, 377, 445, 446,
> 473, 475, 485, 607, 617, 619, 634, 635, 637, 638, 697, 698,
> 701, 725, 727 (not in OEIS)
> > The number of elements up to n should be O(n^k) with k
> = log(2)/log(3) + log(3)/log(4) - 1, approximately k = 0.42.
>  Up to 10^13 the constant is around 3.
> >
> > [
> > Further conjectures:
> > Numbers without a zero in all bases 3..10: infinite
> > Numbers without a zero in all bases 3..15:
> largest=17392214961514563152363
> > Numbers without a zero in all prime bases
> 3<=p<=10^10: infinite
> > Numbers without a zero in all prime bases
> 3<=p<=10^100: finite
> > The behaviour is controlled by the sign of:
> > k = 1-sum{bases b}(1-log(b-1)/log(b))
> > Any help estimating this function for odd primes?
> > ]
> >
> > A,B = palindromes in base a,b
> > Are palindromes k-automatic?
> > I conjecture that any pair of bases that are not
> powers of the same integer give rise to an infinite
> sequence.  The distribution is O(log(n)) in each case.
> > Examples in OEIS: bases 2&10 A007632 (& see
> links), bases 2&3 A060792, bases 3&4 to 6&7
> A097928 to A097931, bases 7&8 A099145, bases 8&9
> A099146.
> >
> > Have I missed something?
> >
> > Martin Fuller
> >
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> >
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/

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